Let $A$ be a normal ring (in the sense that its localizations at prime ideals are normal domains), and suppose that a finite group $G$ acts on $A$ by ring automorphisms. Form the subring $A^G \subset ...

There is a basic theorem in the geometry of schemes saying that the Spec of a Noetherian ring is a Noetherian topological space. It can be formulated as the ACC condition implies the ACCR condition ...

Let $\mathcal{K}$ be a $pointed$ rational cone in $\mathbb{R}^d$ with extremal rays generated by $r_1,r_2,\dots, r_m\in \mathbb{Z}^d$.
Here, pointed means that all $r_i$ lie strictly on one side of ...

Let $R$ be a complete regular local ring whose residue field is perfect. Suppose that a finite group $G$ acts on $R$ by ring automorphisms in such a way that the induced action on the residue field is ...

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...

Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? I mean a statement such as "the given equation holds if and only if (some $\operatorname{Tor}$, ...

Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...

All rings are assumed to be commutative with 1.
We say a module over a ring is strictly noetherian if every non-empty set of submodules has a maximal member. We say a ring is strictly noetherian if it ...

I'm trying to prove some stuff (it's not important what) about (graded) Artinian complete intersections $R=\mathbb{C}[x_1,\ldots,x_n]/I$, where the $x_i$ have certain positive weights and where $I$ is ...

Assume we are blowing up an ideal $I$ on an affine variety $X$, let $E$ be the exceptional divisor, and $P$ be a (closed) point in $V$, the zero set of $I$. Is there any algorithm to check that $E$ ...

Given an ideal $I$ of $\mathbb{R}[X_1,X_2,X_3,X_4,X_5]$ generated by two unknown polynomials. I know two homogenous polynomials $p_1 \in I$ and $p_2 \in I$ such that
$p_1$ is of degree 2 and up to a ...

Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a ...

$(R,m)$ is a local Noetherian ring. $M$ is a nonzero finite $R$-module of finite injective dimension($id$). It is known that if $R$ is Gorenstein, then $M$ has finite flat dimension ($fd$). I wonder ...

Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of ...

I have a problem on Ring theory. I would like to prove or disprove the following statement:
Let $R$ be a Noetherian domain. Then by the Goldie theorem $R$ have $Q$ as a full ring of quotients and $Q$ ...

Let $A$ be a regular ring and $\mathfrak q$ be an ideal, such that $\sqrt{\mathfrak q}$ is prime. Further assume that $\mathfrak q$ is locally principal (i.e. $\mathfrak q$ is an irreducible divisor ...

I am trying to rewrite the theory of decomposition/inertia/ramification groups independently of the theory of Dedekind or valuation rings (I believe this has been done elsewhere, but I found only few ...

(Naive formulation:) Let $X$ be an (irreducible) affine variety (over an algebraically closed field $k$) and $I$ be an ideal of the coordinate ring $R$ of $X$. Assume $Y = V(I)$ is equidimensional. ...